Why can the Bellman equation be turned into an update rule?
Why are we allowed to convert the Bellman equations into update rules?


There is a simple reason for this: convergence. The same chapter 4 of the same book mentions it. For example, in the case of policy evaluation, the produced sequence of estimates $\{v_k\}$ is guaranteed to converge to $v_\pi$ as $k$ (i.e. the number of iterations) goes to infinity. There are other RL algorithms that are also guaranteed to converge (e.g. tabular Q-learning). 

To conclude, in many cases, the update rules of simple reinforcement learning (or dynamic programming) algorithms are very similar to their mathematical formalization because algorithms based on those update rules are often guaranteed to converge. However, note that many more advanced reinforcement learning algorithms (especially, the ones that use function approximators, such as neural networks, to represent the value functions or policies) are not guaranteed or known to converge.